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The arithmetic of certain del Pezzo surfaces and \(K3\) surfaces. (English. French summary) Zbl 1268.14020
It is a classical result of Hasse that smooth quartic hypersurfaces always satisfy the Hasse principle. A degree 4 del Pezzo surface is the smooth intersection of two quadratic hypersurfaces of dimension 4. The author constructs degree 4 del Pezzo surfaces for which the Hasse principle fails because of the Brauer-Manin obstruction. They give explicit equations for the two quadric hypersurfaces defining the surface. One of the surfaces provided has been studied by B. J. Birch and H. P. F. Swinnerton-Dyer [J. Reine Angew. Math. 274–275, 164–174 (1975; Zbl 0326.14007)]. The author then uses the degree 4 del Pezzo surfaces to construct a family of \(K3\) surfaces violating the Hasse principle also. These surfaces are double covers of the del Pezzo branched over a smooth conic.

MSC:
14G05 Rational points
14J28 \(K3\) surfaces and Enriques surfaces
14J26 Rational and ruled surfaces
11G35 Varieties over global fields
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